The most troublesome subexpression is $\sin(69)$, for the $\tan$ continued
fraction expansion behaves poorly, and the terms in the Taylor series
for $\sin(69)$ about 0 grow for quite some time before shrinking to
acceptable levels for continued fractions.

I followed this route in a test program for
the frac library. Starting from $\cos 1$,
I used double-angle formulas to find $\cos 64$, and the fifth Chebyshev
polynomial (of the first kind) to find $\cos 5$. Then since we may square
root continued fractions, we can use $\sin x = \sqrt{1 - \cos ^2 x}$ to find
$\sin 64$ and $\sin 5$. Lastly, the angle addition formula yields $\sin 69$.

Thus using techniques Gosper describes, and the Taylor series, we
find the first 100 decimal places of the HAKMEM constant:

This is gratifying, especially as I could easily optimize further. (For
example, I should be able to remove three integer divisions in my quadratic
algorithm implementation; better ways of finding $\sin 69$ exist).